Everything you've always wanted to know about Hot-S22 (but we're afraid to ask)

Transcription

1 Jan Verspecht bvba Gertrudeveld Steenhuffel Belgium web: Everything you've always wanted to know about Hot-S22 (but we're afraid to ask) Jan Verspecht Presented at the Workshop Introducing New Concepts in Nonlinear Network Design (International Microwave Symposium 2002) 2002 Agilent Technologies - Used with Permission

2 Everything you ve always wanted to know about Hot-S 22 (but were afraid to ask) Jan Verspecht Agilent Technologies 1

3 Purpose Convince people of a better Hot S 22 Show that technology is fun (sometimes) 2 The purpose of this presentation is to convince people that one needs to extend the classic concept of Hot S22 if one wants to describe accurately how a device behaves under large-signal excitation with a non perfect termination. A second purpose is to show that, when the proper set of experiments are performed, Hot S22 can be fun. 2

4 Outline Introduction: What is Hot S 22? Getting and interpreting experimental data Confront classic approaches with data Derivation of the extended Hot S 22 theory Confront extended Hot S 22 with data Conclusion 3 Since there is a lot of confusion on Hot S22 I will first define what I understand by it. Next I will explain how to get and interpret practical experimental data. Then I will confront classical approaches to Hot S22 with this experimental data and I will show that the classic approach is inaccurate. An accurate and extended Hot S22 is then derived from the gathered experimental data. I will show that the new Hot S22 model can indeed accurately describe the measured data, in contrast with the classic approach. I will finally draw conclusions. 3

5 What is Hot S 22? A 1 A 2 D.U.T. B 1 B 2 D.U.T. behavior is represented by pseudo-waves (A 1, B 1, A 2, B 2 ) Hot S 22 describes the relationship between B 2 and A 2 Valid under Hot conditions (A 1 significant) 4 So what do I mean by Hot S22? Or a better question is what problem do we want Hot S22 to solve? The answer is that one wants to describe how a device-under-test (D.U.T.) behaves under variable load conditions at the output, while a large-signal excitation is present. Equivalent to classic linear S-parameters, Hot S22 should solve this problem by describing the relationship between the B2 and the A2 traveling voltage (pseudo-)waves. The important difference with classic S-parameters is that the mathematical model should be valid while a large input signal (A1) is present. This large signal at the input causes the DUT to start behaving in a nonlinear way, such that a classical S-parameter, which is based on the superposition principle, is no longer valid. For the same reason this kind of behavior can not be characterized by a standard commercial network analyzer. This measurement instrument does not only uses the superposition principle for its experiment design (one-tone applied at the same time at one of the DUT ports), but also for its advanced and sophisticated calibration procedures. 4

6 Experimental investigation Take a real life D.U.T. (CDMA RFIC amplifier) Apply an A 1 signal Apply a set of A 2 s Look at the corresponding B 2 s Mathematically describe the relationship between the A 2 s and B 2 s Repeat for different A 1 s 5 The approach used in this presentation is to come up with an accurate Hot S22 measurement method and mathematical model based upon observations of DUT behavior. In order to do this the following experiment is performed. First one takes a real life DUT. In our case this is an RFIC power amplifier aimed for CDMA applications. Next one applies a largesignal at the input, this implies the application of a significant A1 spectral component at the input of the RFIC. Next one applies an extended set of A2 waves. These are generated by a second synthesizer and are injected towards the RFIC output. Then one records for all applied A2 s the corresponding B2 s and one tries to discover the mathematical relationship between the A2 s and the B2 s. Finally one repeats this process for different A1 s (typically one performs a power sweep). The mathematical relationship that is discovered between the A2 s and the B2 s is what we call the Hot S22 -model of the DUT. 5

7 Experimental set-up Large-Signal Network Analyzer (LSNA, formerly known as NNMS) synth1 synth2 RFIC A 1 B 1 B 2 A 2 synth1 generates A 1 synth2 generates a set of A 2 s LSNA measures all A 1 s, B 1 s, A 2 s, B 2 s 6 The experimental set-up we used looks as follows. The set-up can be viewed in some sense as an extended vector network analyzer. The set-up contains two synthesizers synth1 and synth2. Synth1 generates the large-signal input, noted A1. This ensures that all measurements can be performed under hot conditions. Synth2 generates a set of A2 s. For a good experiment design it is very important that A2 can be applied independent from A1. The two synthesizers can put the DUT in all experimental conditions that are needed to investigate Hot S22. Note that it is possible to use a tuner in stead of a Synth2. In our case the use of a synthesizer makes the set-up more flexible. The synthesizer allows, for example, to emulate any given load impedance through active loadpull. Next to the signal generation, there is of course the problem of the accurate measurement of the A1 s, the A2 s and the B2 s. A simple vector network analyzer is of no use here since it can merely measure ratio s of waves (the S-parameters) rather than the absolute waves. For the purpose of absolutely measuring the waves we connect the DUT to the test-set of a Large-Signal Network Analyzer, the instrument formerly known as NNMS. This instrument allows to accurately detect the incident and scattered (pseudo-)voltage waves A1, B1, A2 and B2 which appear at both of the DUT signal ports. 6

8 Interpretation of the data A 2 (V) B 2 (V) ?0.1?0.2?0.3?0.3?0.2? ?0.25?0.5?0.75?0.75? 0.5? IQ-plots of the A 2 s and B 2 s for a constant A 1 (x-axis = real part, y-axis = imaginary part) 7 Next we do the following experiment. We choose one value for A1. Then we apply a set of different A2 s and we look at the relationship between the B2 s and the A2 s for the particular value of A1. In the graph we have plotted the set of applied A2 s and the corresponding set of B2 s in so-called IQ-plots. This means that we plot the real part of each measured complex number on the x-axis and the corresponding imaginary part on the y-axis. At a first glance it is hard to interpret the data. In the past people have turned to Volterra and other theories in order to get the mathematical relationship between the two measured quantities A2 and B2. An simple intuitive interpretation is possible, however, by introducing a so-called phase normalization. The idea is to use the A1 wave as a phase reference. This is done as follows. Suppose that one has one measurement of the quantities A1, B2 and A2. One will then apply a phase shift to all of these quantities which equals the opposite of the phase of A1. As a result one gets a new set of A1, A2 and B2 where A1 has a zero phase. In what follows this kind of phase normalization is always used for representing A2 and B2. 7

9 Phase normalization is good ?0.1?0.2?0.3 A 2.P -1 (V)?0.3? 0.2? B 2.P -1 (V)?0.6?0.4?0.2 0 Normalize the phases relative to A 1 P = e j.arg(a1) 8 When applied to the data previously shown, it is clear that the result leans itself better for an intuitive understanding of the relationship between the A2 s and the B2 s. Note that the phase normalization is mathematically expressed as a division by the complex phasor P, which has a phase equal to the phase of A1 but which has a unity amplitude. 8

10 Varying the amplitude of A 1 B 2.P -1 (V) A1 increases ?1.25?1?0.75?0.5? Let us now do an experiment, using the same set of A2 s, but varying the amplitude of A1. If we plot the set of corresponding B2 s we note several things. First of all the midpoint of the resulting B2 s changes and gets a larger amplitude with increasing A1. This makes a lot of sense. The midpoint of the B2 s can be interpreted as the output of the amplifier when it is perfectly matched (A2 equal to zero). Applying the A2 s can be viewed as applying a deviation from the perfect match. We also note that these midpoints are approximately located on a straight line. This shows that the component that we are measuring has insignificant AM-to-PM. But the most striking observation is that the smiley gets distorted as A1 increases. The question now is whether a classic Hot S22 predicts this kind of behavior, and, if this is not the case, whether it is possible to come up with an alternative solution. 9

11 Varying the amplitude of A 2?1.3?1.2?1.1?1.3?1.2?1.1?1.3?1.2? B 2.P -1 (V) ?1.3?1.2?1.1?1.3?1.2?1.1?1.3?1.2?1.1?0.5? ?0.5? ?0.5? A 2.P -1 (V) 0.2 0? ? ?0.2?0.4?0.4?0.4?0.6?0.6?0.6?0.5? ?0.5? ?0.5? Linear dependency versus A 2 10 Next we will do another experiment. We will now keep A1 constant and we will vary the amplitude of A2. From the plots above we can conclude that, in a good approximation and despite the distortion, the B2 deviation from the midpoint is proportional to A2. The brings us to the conclusion that the relationship is almost linear. 10

12 Data interpretation as loadpull Increasing amplitude of A 1 11 Just out of curiosity, the experimental data can also be viewed in a loadpull mode. The graph shown corresponds to the experiment with increasing amplitude of A1 (the set of A2 s is kept constant). For a loadpull graph one plots the ratio of A2 and B2 on a Smith chart in order to see the corresponding reflection coefficients and impedances. Note that the area covered by the smiley decreases for an increasing level of A1. This is because the corresponding B2 have a higher amplitude. 11

13 Classic S-parameter description B 2.P -1 (V) ?1.25?1?0.75? 0.5? B 2 = S 21 ( A 1 ).A 1 + S 22.A 2 12 Let us know check how good existing approaches can explain the experimental data corresponding to the increasing amplitude of A1. As a first approach I consider a classic S22-parameter, combined with a large-signal S21, which in fact corresponds to a compression and AM-PM characteristic. The mathematical relationship between the B2 s and the A2 s and A1 s in this case is given by a simple S-parameter relationship, where the S21 has become a function of the amplitude of A1. The experimental data is depicted in orange, the S22-model is depicted in yellow. As we expected the S22 is not capable of describing the fact that the relationship between the A2 s and B2 s is clearly a function of the amplitude of A1. This S22-model is also not capable of describing the kind of distortion that we see in the experimental data. 12

14 Simple Hot S 22 description B 2.P -1 (V) ?1.25?1?0.75? 0.5? B 2 = S 21 ( A 1 ).A 1 + S 22 ( A 1 ).A 2 13 In the past, people improved the accuracy of the previous model by also making S22 a function of the amplitude of A1. The resulting mathematical equation is shown above. This Hot S22 can actually be measured by modified vector network analyzer setups. It is clear that this model is linear in A2 (what we want it to be), and can handle a varying amplitude of A1. Unfortunately this kind of model can never describe the kind of distortion that we see in the experimental data. It still looks like something is missing. 13

15 Model linearity & squeezing We look for a mathematical model which is linear (superposition valid) squeezes Squeezing implies that the phase of A 2.P -1 matters We need different coefficients for the real and the imaginary part of A 2.P -1 More elegant expression results when using A 2.P -1 and its conjugate 14 So we are looking for an alternative model which: -Is linear in A2 -Describes the squeezing of the smiley -Is a general function of the amplitude of A1 The fact that we see a flattening of the smiley implies that the phase of A2 relative to A1 matters. This implies that we should use a different coefficient for the real and imaginary part of the phase normalized A2. This is in fact equivalent to using a different coefficient for the phase normalized A2 and its conjugate (this results in a more elegant expression). 14

16 Mathematical expression B 2.P -1 = S 21 ( A 1 ).A 1.P -1 + S 22 ( A 1 ).A 2.P -1 + R 22 ( A 1 ).conjugate(a 2.P -1 ) B 2 = S 21 ( A 1 ).A 1 + S 22 ( A 1 ).A 2 + R 22 ( A 1 ).P 2.conjugate(A 2 ) 15 The resulting expression is shown above. We start by writing a linear relationship between the phase normalized quantities B2, A2, the conjugate of A2, and A1, where each of the coefficients is a general function of the amplitude of A1. Next we multiply both sides by the phasor P, which results in the lower expression. This expression is actually an extension of the previous Hot S22, where a linear term is added including the conjugate of A2. Note that this variable is always multiplied by the square of P. As such this term is not only a function of the amplitude of A1 but also of the phase between A2 and A1. 15

17 Extended Hot S 22 B 2.P -1 (V) ?1.25?1?0.75? 0.5? B 2 = S 21 ( A 1 ).A 1 + S 22 ( A 1 ).A 2 + R 22 ( A 1 ).P 2.conjugate(A 2 ) 16 We will now confront the so-called extended Hot S22 with the experimental data. We immediately see that the introduction of the conjugate term is precisely what is needed in order to describe the squeezing that takes place. Note that the resulting model is still linear in A2! 16

18 Quadratic Hot S 22 Further improvement is possible by using a polynomial in A 2 and conj(a 2 ) E.g.: quadratic Hot S 22 B 2 = F.P + G.A 2 + H.P 2.conj(A 2 ) + K.P -1.A L.P 3.conj(A 2 ) 2 + M.P.A 2.conj(A 2 ) Note the presence of the P factors (theory of describing functions) 17 For those who want even more accuracy, especially for an increasing amplitude of A2, the idea can easily be extended to describe nonlinear dependencies on A2. The result is shown above. Although not explicitly written for clarity, all coefficients F, G, H, K, L and M are general functions of the amplitude of A1. Also note the presence of the P factors raised to a certain power (factors given by the theory of describing functions ). 17

19 Comparison (highest A 1 amplitude) B 2.P -1 (V)?1.4?1.3?1.2?1.4?1.3?1.2 Classic S Simple Hot S22?1.4?1.3?1.2?1.4?1.3?1.2?1.4?1.3?1.2?1.4?1.3?1.2 Extended Hot S Quadratic Hot S ?1.4?1.3?1.2?1.4?1.3? The figure above shows the performance of the 4 Hot S22 approaches when confronted with the experimental data. A classic S22 is clearly inaccurate since it is not a function of A1. The simple Hot S22 includes the dependency on the amplitude of A1. The extended Hot S22 also includes the dependency on the phase of A2 versus A1 ( the squeeze ), resulting in a significantly better match between experiment and model. An even better match results with the quadratic Hot S22, which describes a 2 nd order nonlinear dependency on A2. 18

20 Residuals (db)?30?35?40 Relative Error S 22 Simple Hot S 22?45?50?25?20?15?10?5 0 5 Amplitude of A 1 (dbm) Extended Hot S 22 Quadratic Hot S A more qualitative measure for the performance of the models is seen when looking at their residuals (relative to the total energy in the signal) as a function of the amplitude of A1. It is clear from the graph that the addition of the conjugate term significantly reduces the value of the residual (up to 15 db), except for the lowest value of A1. This makes a lot of sense since for low amplitudes of A1 the Hot S22 reduces to a perfectly linear S- parameter model which contains no conjugate term (the same is true for the quadratic Hot S22 ). We also see that the quadratic Hot S22, for our measurements, only results in a marginal improvement (3 db), and this only for the highest input powers of A1. Note that the relative residual becomes lower than 50 db with the proposed extended Hot S22, which approaches the dynamic range of our measurement system (about 60 db for this set of measurements). 19

21 Conclusion An accurate Hot S 22 exists It has a coefficient for the conjugate of A 2.P -1 It can accurately be measured It describes the relationship between A 2 and B 2 under large-signal excitation 20 There exists a more accurate Hot S22 concept. It contains a linear term in the conjugate of the phase normalized A2. It can be measured with a Large-Signal Network Analyzer and accurately describes the relationship between B2 and A2 under a large signal excitation. 20

22 More information More detailed information on this kind of measuring and modeling techniques: